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Chapter 2 Binary Values and Number Systems Numbers Natural Numbers Zero and any number obtained by repeatedly adding one to it. Examples: 100, 0, 45645, 32 Negative Numbers A value less than 0, with a – sign Examples: -24, -1, -45645, -32 2 2 Numbers Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 Rational Numbers An integer or the quotient of two integers Examples: -249, -1, 0, 3/7, -2/5 3 3 Natural Numbers How many ones are there in 642? 600 + 40 + 2 ? Or is it 384 + 32 + 2 ? Or maybe… 1536 + 64 + 2 ? 4 4 Natural Numbers Aha! 642 is 600 + 40 + 2 in BASE 10 The base of a number determines the number of digits and the value of digit positions 5 5 Positional Notation Continuing with our example… 642 in base 10 positional notation is: 6 x 102 = 6 x 100 = 600 + 4 x 101 = 4 x 10 = 40 + 2 x 10º = 2 x 1 = 2 = 642 in base 10 This number is in base 10 6 The power indicates the position of the number 6 Positional Notation Example: 642 is 6 * 102 + 4 * 10 + 2 R is the base of the number As a formula: dn * Rn-1 + dn-1 * Rn-2 + ... + d2 * R + d1 n is the number of digits in the number 7 d is the digit in the ith position in the number 7 Positional Notation What if 642 has the base of 13? + 6 x 132 = 6 x 169 = 1014 + 4 x 131 = 4 x 13 = 52 + 2 x 13º = 2 x 1 = 2 = 1068 in base 10 642 in base 13 is equivalent to 1068 in base 10 8 8 6 Binary Decimal is base 10 and has 10 digits: 0,1,2,3,4,5,6,7,8,9 Binary is base 2 and has 2 digits: 0,1 For a number to exist in a given base, it can only contain the digits in that base, which range from 0 up to (but not including) the base. What bases can these numbers be in? 122, 198, 178 9 9 Bases Higher than 10 How are digits in bases higher than 10 represented? With distinct symbols for 10 and above. Base 16 has 16 digits: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E, and F 10 10 Converting Octal to Decimal What is the decimal equivalent of the octal number 642? 6 x 82 = 6 x 64 = 384 + 4 x 81 = 4 x 8 = 32 + 2 x 8º = 2 x 1 = 2 = 418 in base 10 11 11 Converting Hexadecimal to Decimal What is the decimal equivalent of the hexadecimal number DEF? (Remember, the digits in base 16 are 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F) D x 162 = 13 x 256 = 3328 + E x 161 = 14 x 16 = 224 + F x 16º = 15 x 1 = 15 = 3567 in base 10 12 Converting Binary to Decimal What is the decimal equivalent of the binary number 1101110? 1 x 26 + 1 x 25 + 0 x 24 + 1 x 23 + 1 x 22 + 1 x 21 + 0 x 2º 13 = = = = = = = 1 x 64 1 x 32 0 x 16 1x8 1x4 1x2 0x1 = 64 = 32 =0 =8 =4 =2 =0 = 110 in base 10 13 Arithmetic in Decimal How do we add 2 decimal numbers? Example: 197 + 265 = ? 14 Arithmetic in Binary Remember that there are only 2 digits in binary, 0 and 1 1 + 1 is 0 with a carry 111111 1010111 +1 0 0 1 0 1 1 10100010 15 Carry Values 14 Subtracting Decimal Numbers Anybody remember how to borrow in order to subtract two numbers? Example: 16 271 – 196 = ? Subtracting Binary Numbers Apply that same concept here: 12 202 1010111 - 111011 0011100 17 15 Counting in Binary/Octal/Decimal 18 A Trick: Converting Binary to Octal • Mark groups of three (from right) • Convert each group 10101011 10 101 011 2 5 3 10101011 is 253 in base 8 Why does this work?? 19 17 Converting Binary to Hexadecimal • Mark groups of four (from right) • Convert each group 10101011 1010 1011 A B 10101011 is AB in base 16 And this? 20 18 Converting Decimal to Other Bases Algorithm for converting number in base 10 to other bases While (the quotient is not zero) Divide the decimal number by the new base Make the remainder the next digit to the left in the answer Replace the original decimal number with the quotient 21 19 Converting Decimal to Octal What is 1988 (base 10) in base 8? Try it! 22 Converting Decimal to Octal 248 8 1988 16 38 32 68 64 4 31 8 248 24 08 8 0 3 8 31 24 7 Answer is : 3 7 0 4 23 0 8 3 0 3 Converting Decimal to Hexadecimal What is 3567 (base 10) in base 16? Try it! 24 20 Converting Decimal to Hexadecimal 222 16 3567 32 36 32 47 32 15 13 16 222 16 62 48 14 0 16 13 0 13 DEF 25 21 Binary Numbers and Computers Computers have storage units called binary digits or bits Low Voltage = 0 High Voltage = 1 26 all bits have 0 or 1 22 Binary and Computers Byte 8 bits The number of bits in a word determines the word length of the computer, but it is usually a multiple of 8 • 32-bit machines • 64-bit machines etc. 27 23